TY - JOUR
T1 - Stochastic mode reduction for the immersed boundary method
AU - Kramer, Peter R.
AU - Majda, Andrew J.
N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.
PY - 2004
Y1 - 2004
N2 - We apply the formulation of a stochastic mode reduction method developed in a recent paper of Majda, Timofeyev, and Vanden-Eijnden [Comm. Pure Appl. Math., 54 (2001), pp. 891-974] (MTV) to obtain simplified equations for the dynamics of structures immersed in a thermally fluctuating fluid at low Reynolds (or Kubo) number, as simulated by a recent extension of the immersed boundary (IB) method by Kramer and Peskin [Proceedings of the Second MIT Conference on Computational Fluid and Solid Mechanics, Elsevier Science, Oxford, UK, 2003, pp. 1755-1758]. The effective dynamics of the immersed structures are not obvious in the primitive equations, which involve both fluid and structure dynamics, but the procedure of MTV allows the rigorous derivation of a reduced stochastic system for the immersed structures alone. We find, in the limit of small Reynolds (or Kubo) number, that the Lagrengian particle constituents of the immersed structures undergo a drift-diffusive motion with several physically correct features, including the coupling between dynamics of different particles. The MTV procedure is also applied to the spatially discretized form of the IB equations with thermal fluctuations to assist in the design and assessment of numerical algorithms.
AB - We apply the formulation of a stochastic mode reduction method developed in a recent paper of Majda, Timofeyev, and Vanden-Eijnden [Comm. Pure Appl. Math., 54 (2001), pp. 891-974] (MTV) to obtain simplified equations for the dynamics of structures immersed in a thermally fluctuating fluid at low Reynolds (or Kubo) number, as simulated by a recent extension of the immersed boundary (IB) method by Kramer and Peskin [Proceedings of the Second MIT Conference on Computational Fluid and Solid Mechanics, Elsevier Science, Oxford, UK, 2003, pp. 1755-1758]. The effective dynamics of the immersed structures are not obvious in the primitive equations, which involve both fluid and structure dynamics, but the procedure of MTV allows the rigorous derivation of a reduced stochastic system for the immersed structures alone. We find, in the limit of small Reynolds (or Kubo) number, that the Lagrengian particle constituents of the immersed structures undergo a drift-diffusive motion with several physically correct features, including the coupling between dynamics of different particles. The MTV procedure is also applied to the spatially discretized form of the IB equations with thermal fluctuations to assist in the design and assessment of numerical algorithms.
KW - Brownian motion
KW - Immersed boundary method
KW - Stochastic mode reduction
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U2 - 10.1137/S0036139903422139
DO - 10.1137/S0036139903422139
M3 - Article
AN - SCOPUS:2542506478
VL - 64
SP - 369
EP - 400
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
SN - 0036-1399
IS - 2
ER -